Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system of nonlinear equations using substitution. $$ \begin{array}{r} y=-x \\ x^{2}+y^{2}=9 \end{array} $$

Step-by-Step Solution

Answer

Solutions are \( \left( \sqrt{\frac{9}{2}}, -\sqrt{\frac{9}{2}} \right) \) and \( \left( -\sqrt{\frac{9}{2}}, \sqrt{\frac{9}{2}} \right) \).

Step 1: Understand the System of Equations

We have two equations to work with. The first equation is a linear equation, \( y = -x \). The second equation is a circle equation \( x^2 + y^2 = 9 \). We will use substitution to solve this system.

Step 2: Substitute y in the Circle Equation

Using the first equation \( y = -x \), plug this into the second equation. The second equation \( x^2 + y^2 = 9 \) becomes: \( x^2 + (-x)^2 = 9 \). This simplifies to \( x^2 + x^2 = 9 \).

Step 3: Simplify and Solve for x

Combine like terms in the equation \( x^2 + x^2 = 9 \) to get \( 2x^2 = 9 \). Divide both sides by 2 to obtain \( x^2 = \frac{9}{2} \). Take the square root of both sides to solve for \( x \): \( x = \pm \sqrt{\frac{9}{2}} \).

Step 4: Solve for y Using x Values

Substitute \( x = \sqrt{\frac{9}{2}} \) into the equation \( y = -x \) to find \( y = -\sqrt{\frac{9}{2}} \). Substitute \( x = -\sqrt{\frac{9}{2}} \) into \( y = -x \) to find \( y = \sqrt{\frac{9}{2}} \).

Step 5: Write the Solution

The solutions to the system of equations are \( (x, y) = \left( \sqrt{\frac{9}{2}}, -\sqrt{\frac{9}{2}} \right) \) and \( (x, y) = \left( -\sqrt{\frac{9}{2}}, \sqrt{\frac{9}{2}} \right) \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used to solve systems of equations, especially useful when dealing with nonlinear equations. It involves replacing one variable with an expression obtained from another equation in the system. This technique helps to simplify the process by reducing the number of variables involved, turning a system of equations into a single equation that is easier to solve.
To apply the substitution method, follow these basic steps:

• Start by solving one of the equations for one variable. In the given problem, we have the simple equation \( y = -x \).
• Use the solved expression to substitute in place of that variable in the other equation. Here, substitute \( y = -x \) into the circle equation \( x^2 + y^2 = 9 \).
• Simplify and solve the new equation for the remaining variable.
• Find the value of the second variable by substituting back the solution obtained into the expression found in the first step.

This method is especially friendly for nonlinear systems, where equations might include squares or other non-linear terms, as it turns them into simpler linear equations.

Circle Equation

A circle equation typically comes in the form \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle, and \( (x, y) \) are the coordinates of any point on the circle. In the context of solving systems of equations, the circle equation is key, as it sets the path along which solutions must lie.
In our exercise, \( x^2 + y^2 = 9 \) represents a circle centered at the origin \((0,0)\) with a radius of 3, since \( 9 = 3^2 \). All points that satisfy this equation are exactly 3 units away from the origin.
Understanding the geometric representation can aid in visualizing the problem. Since we are also solving the equation \( y = -x \), which is a line, the solutions are the points where this line intersects the circle. These intersection points are the solutions to both equations simultaneously, illustrating the concept of solving systems of equations geometrically.

Solving Systems of Equations

Solving systems of equations involves finding values of variables that satisfy all equations in the system simultaneously. The system can be linear, nonlinear, or a mix of both, as is the case in the exercise provided. Different methods can be used, including:

• Graphical Method: Plotting each equation on a graph to find intersection points, appropriate for systems with visual components.
• Substitution Method: As discussed, involves substituting one equation into another to reduce the system to a single equation, ideal for systems with an easy solvable equation.
• Elimination Method: Involves adding or subtracting equations to eliminate a variable. Works well with linear systems but can be applied to certain nonlinear systems.

In the exercise, substitution is ideal because the equation \( y = -x \) easily substitutes into the circle equation, reducing it to a form that's more straightforward to solve. The solution showcases pairs \((x, y)\) that satisfy both the linear line and the circle equation, emphasizing the intersection concept inherent in solving systems of equations.